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We derive some analytic closed-form solutions for a class of Riccati equation y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are C^{\infty}-functions. We show that if \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0,…
We revisit and extend the Riccati theory, unifying continuous-time linear-quadratic optimal permanent and sampled-data control problems, in finite and infinite time horizons. In a nutshell, we prove that:-- when the time horizon T tends to…
Robust optimization is a framework for modeling optimization problems involving data uncertainty and during the last decades has been an area of active research. If we focus on linear programming (LP) problems with i) uncertain data, ii)…
The algebraic intersection type unification problem is an important component in proof search related to several natural decision problems in intersection type systems. It is unknown and remains open whether the algebraic intersection type…
We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately,…
The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither $H^2$-regularity nor $ L^{\infty} $-error estimation, but only $ H^1_0…
We present several second-order linear differential equations that are associated to a particular Riccati equation with only one constant parameter in its coefficients through the technique of supersymmetric factorizations and through a…
We construct singular solutions of a complex elliptic equation of second order, having an isolated singularity of any order. In particular, we extend results obtained for the real partial differential equation in divergence form by…
Discrete-time robust optimal control problems generally take a min-max structure over continuous variable spaces, which can be difficult to solve in practice. In this paper, we extend the class of such problems that can be solved through a…
We present the first accelerated randomized algorithm for solving linear systems in Euclidean spaces. One essential problem of this type is the matrix inversion problem. In particular, our algorithm can be specialized to invert positive…
We establish a well-posedness and error-estimation framework that solves Hamilton-Jacobi equations by minimizing the least-squares residual of monotone finite-difference discretizations. This approach also applies naturally to second-order…
We utilise recent results about the transcendental solutions to Riccati differential equations to provide a comprehensive description of the nature of the transcendental solutions to algebraic first order differential equations of genus…
We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations. For an autonomous first-order algebraic ordinary difference equations, we give an upper bound for the degrees…
Methods from learning theory are used in the state space of linear dynamical and control systems in order to estimate the system matrices. An application to stabilization via algebraic Riccati equations is included. The approach is…
Uniqueness of positive solutions to viscous Hamilton-Jacobi-Bellman (HJB) equations of the form $-\Delta u(x) + \frac{1}{\gamma} |D{u}(x)|^\gamma = f(x) - \lambda$, with $f$ a coercive function and $\lambda$ a constant, in the subquadratic…
Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in [7]. We…
This work is concerned with the numerical solution of large-scale symmetric positive definite matrix equations of the form $A_1XB_1^\top + A_2XB_2^\top + \dots + A_\ell X B_\ell^\top = F$, as they arise from discretized partial differential…
We show that the Riccati form of the Schrodinger equation can be reformulated in terms of two linear equations depending on an arbitrary function G. When $G$ and the potential are polynomials, the solutions of these two equations are entire…
The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete…
The Riccati equation method is used to establish some new stability criteria for systems of two linear first-order ordinary differential equations. It is shown that two of these criteria in the two dimensional case imply the Routh -…